On solutions to the Ginzburg-Landau equations in higher dimensions

TL;DR

This paper proves a new theorem for the Ginzburg-Landau equations in higher dimensions, showing solutions can concentrate near minimal submanifolds of codimension 2 using a sophisticated construction and the implicit function theorem.

Contribution

It establishes a glueing theorem for Ginzburg-Landau equations in dimensions greater than two, linking solutions to minimal submanifolds of codimension 2.

Findings

Solutions concentrate near minimal submanifolds of codimension 2

Constructs a one-parameter family of solutions

Uses approximate solutions and the implicit function theorem

Abstract

We establish a glueing theorem for the Ginzburg-Landau equations in dimension $n > 2$. To this end, we consider a nondegenerate minimal submanifold of codimension 2, and construct a one-parameter family of solutions to the Ginzburg-Landau equations such that the energy density concentrates near this submanifold. The proof is based on a construction of suitable approximate solutions and the implicite function theorem.